Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [315,3,Mod(37,315)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(315, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([0, 3, 4]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("315.37");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 315.ca (of order \(12\), degree \(4\), minimal) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
Self dual: | no |
Analytic conductor: | \(8.58312832735\) |
Analytic rank: | \(0\) |
Dimension: | \(64\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{12})\) |
Twist minimal: | no (minimal twist has level 105) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-expansion
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
Label | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
37.1 | −3.52852 | + | 0.945463i | 0 | 8.09242 | − | 4.67216i | −3.83301 | − | 3.21062i | 0 | −6.80203 | + | 1.65300i | −13.8047 | + | 13.8047i | 0 | 16.5603 | + | 7.70474i | ||||||
37.2 | −3.41166 | + | 0.914152i | 0 | 7.33966 | − | 4.23756i | 4.98672 | − | 0.364163i | 0 | −6.35750 | − | 2.92953i | −11.1766 | + | 11.1766i | 0 | −16.6801 | + | 5.80102i | ||||||
37.3 | −2.88660 | + | 0.773463i | 0 | 4.27013 | − | 2.46536i | −0.160937 | + | 4.99741i | 0 | 5.19830 | − | 4.68804i | −1.96674 | + | 1.96674i | 0 | −3.40075 | − | 14.5500i | ||||||
37.4 | −2.56059 | + | 0.686107i | 0 | 2.62176 | − | 1.51367i | −1.32982 | + | 4.81992i | 0 | 0.158915 | + | 6.99820i | 1.82323 | − | 1.82323i | 0 | 0.0981374 | − | 13.2542i | ||||||
37.5 | −2.13226 | + | 0.571337i | 0 | 0.756006 | − | 0.436480i | −2.78563 | − | 4.15214i | 0 | 2.73668 | − | 6.44287i | 4.88107 | − | 4.88107i | 0 | 8.31195 | + | 7.26192i | ||||||
37.6 | −0.984292 | + | 0.263740i | 0 | −2.56483 | + | 1.48081i | 4.04958 | + | 2.93273i | 0 | −5.81621 | + | 3.89508i | 5.01620 | − | 5.01620i | 0 | −4.75945 | − | 1.81862i | ||||||
37.7 | −0.901161 | + | 0.241465i | 0 | −2.71032 | + | 1.56480i | −2.44472 | − | 4.36158i | 0 | 5.41162 | + | 4.44009i | 4.70337 | − | 4.70337i | 0 | 3.25625 | + | 3.34017i | ||||||
37.8 | −0.232416 | + | 0.0622758i | 0 | −3.41396 | + | 1.97105i | 2.65242 | − | 4.23847i | 0 | −4.06422 | + | 5.69931i | 1.35127 | − | 1.35127i | 0 | −0.352512 | + | 1.15027i | ||||||
37.9 | 0.435396 | − | 0.116664i | 0 | −3.28814 | + | 1.89841i | −3.62365 | + | 3.44517i | 0 | 6.87993 | + | 1.29096i | −2.48509 | + | 2.48509i | 0 | −1.17579 | + | 1.92276i | ||||||
37.10 | 0.445275 | − | 0.119311i | 0 | −3.28007 | + | 1.89375i | −4.59550 | + | 1.97013i | 0 | −0.171680 | − | 6.99789i | −2.53844 | + | 2.53844i | 0 | −1.81120 | + | 1.42554i | ||||||
37.11 | 1.84280 | − | 0.493776i | 0 | −0.312015 | + | 0.180142i | 1.82066 | + | 4.65674i | 0 | −6.63712 | − | 2.22456i | −5.88211 | + | 5.88211i | 0 | 5.65448 | + | 7.68243i | ||||||
37.12 | 1.91023 | − | 0.511845i | 0 | −0.0771041 | + | 0.0445161i | 4.99333 | + | 0.258093i | 0 | 6.99587 | + | 0.240410i | −5.71805 | + | 5.71805i | 0 | 9.67053 | − | 2.06280i | ||||||
37.13 | 2.38023 | − | 0.637781i | 0 | 1.79464 | − | 1.03614i | −2.91424 | + | 4.06291i | 0 | −4.25379 | + | 5.55925i | −3.35897 | + | 3.35897i | 0 | −4.34532 | + | 11.5293i | ||||||
37.14 | 2.87375 | − | 0.770020i | 0 | 4.20143 | − | 2.42570i | −3.78688 | − | 3.26489i | 0 | −1.65502 | − | 6.80154i | 1.79111 | − | 1.79111i | 0 | −13.3966 | − | 6.46653i | ||||||
37.15 | 3.18498 | − | 0.853412i | 0 | 5.95167 | − | 3.43620i | 4.91224 | − | 0.932701i | 0 | 5.00478 | + | 4.89410i | 6.69718 | − | 6.69718i | 0 | 14.8494 | − | 7.16279i | ||||||
37.16 | 3.56483 | − | 0.955194i | 0 | 8.33154 | − | 4.81021i | 1.05941 | − | 4.88648i | 0 | −6.28878 | + | 3.07429i | 14.6673 | − | 14.6673i | 0 | −0.890898 | − | 18.4314i | ||||||
163.1 | −0.955194 | − | 3.56483i | 0 | −8.33154 | + | 4.81021i | 3.70210 | − | 3.36072i | 0 | 3.07429 | + | 6.28878i | 14.6673 | + | 14.6673i | 0 | −15.5166 | − | 9.98725i | ||||||
163.2 | −0.853412 | − | 3.18498i | 0 | −5.95167 | + | 3.43620i | −1.64838 | − | 4.72047i | 0 | 4.89410 | − | 5.00478i | 6.69718 | + | 6.69718i | 0 | −13.6279 | + | 9.27855i | ||||||
163.3 | −0.770020 | − | 2.87375i | 0 | −4.20143 | + | 2.42570i | 4.72092 | + | 1.64709i | 0 | −6.80154 | + | 1.65502i | 1.79111 | + | 1.79111i | 0 | 1.09812 | − | 14.8351i | ||||||
163.4 | −0.637781 | − | 2.38023i | 0 | −1.79464 | + | 1.03614i | −2.06146 | + | 4.55526i | 0 | 5.55925 | + | 4.25379i | −3.35897 | − | 3.35897i | 0 | 12.1573 | + | 2.00150i | ||||||
See all 64 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
7.c | even | 3 | 1 | inner |
35.l | odd | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 315.3.ca.b | 64 | |
3.b | odd | 2 | 1 | 105.3.v.a | ✓ | 64 | |
5.c | odd | 4 | 1 | inner | 315.3.ca.b | 64 | |
7.c | even | 3 | 1 | inner | 315.3.ca.b | 64 | |
15.e | even | 4 | 1 | 105.3.v.a | ✓ | 64 | |
21.h | odd | 6 | 1 | 105.3.v.a | ✓ | 64 | |
35.l | odd | 12 | 1 | inner | 315.3.ca.b | 64 | |
105.x | even | 12 | 1 | 105.3.v.a | ✓ | 64 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
105.3.v.a | ✓ | 64 | 3.b | odd | 2 | 1 | |
105.3.v.a | ✓ | 64 | 15.e | even | 4 | 1 | |
105.3.v.a | ✓ | 64 | 21.h | odd | 6 | 1 | |
105.3.v.a | ✓ | 64 | 105.x | even | 12 | 1 | |
315.3.ca.b | 64 | 1.a | even | 1 | 1 | trivial | |
315.3.ca.b | 64 | 5.c | odd | 4 | 1 | inner | |
315.3.ca.b | 64 | 7.c | even | 3 | 1 | inner | |
315.3.ca.b | 64 | 35.l | odd | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{64} + 8 T_{2}^{61} - 468 T_{2}^{60} - 48 T_{2}^{59} + 32 T_{2}^{58} - 3248 T_{2}^{57} + \cdots + 169835630410000 \) acting on \(S_{3}^{\mathrm{new}}(315, [\chi])\).